Published: Journal of Risk, Fall 1998
INCORPORATING VOLATILITY UPDATING INTO THE HISTORICAL
SIMULATION METHOD FOR VALUE AT RISK
John Hull and Alan White
University of Toronto
First Draft: March 1998; This Version : October 1998
ABSTRACT
This paper proposes a procedure for using a GARCH or exponentially weighted moving
average model in conjunction with historical simulation when computing value at risk. It
involves adjusting historical data on each market variable to reflect the difference between
the historical volatility of the market variable and its current volatility. We compare the
approach using nine years of daily data on 12 exchange rates and 5 stock indices with the
historical simulation approach and show that it is a substantial improvement.
INCORPORATING VOLATILITY UPDATING INTO THE HISTORICAL
SIMULATION METHOD FOR VALUE AT RISK
John Hull and Alan White1
University of Toronto
1. Introduction
In recent years value at risk (VaR) has become a very popular measure of market risk. It is
widely used by financial institutions, fund managers, and nonfinancial corporations to
control the market risk in a portfolio of financial instruments. As discussed by Jorion
(1997), it has been adopted by central bank regulators as the major determinant of the
capital banks are required to keep to cover potential losses arising from the market risks
they are bearing.
The VaR of a portfolio is a function of two parameters, a time period and a confidence
level. It equals the dollar loss on the portfolio that will not be exceeded by the end of the
time period with the specified confidence level. If X% is the confidence level and N days is
the time period, the calculation of VaR is based on the probability distribution of changes
in the portfolio value over N days. Specifically VaR is set equal to the loss on the portfolio
at the 100-X percentile point of the distribution. Bank regulators have chosen N equal to
10 days and X equal to 99%. They set the capital required for market risk equal to three
times the value of VaR calculated using these parameters.
In practice the VaR for N days is almost invariably assumed to be √N times the VaR for
one day. A key task for risk managers has therefore been the development of accurate and
robust procedures for calculating a one-day VaR.
One common approach to calculating VaR involves assuming that daily percentage
changes in the underlying market variables are conditionally multivariate normal with the
mean percentage change in each market variable being zero. This is often referred to as the
“model building” approach. If the daily change in the portfolio value is linearly dependent
on daily changes in market variables that are normally distributed, its probability
distribution is also normal. The variance of the probability distribution, and hence the
percentile of the distribution corresponding to VaR, can be calculated in a straightforward
way from the variance-covariance matrix for the market variables. In circumstances where
the linear assumption is inappropriate, the change in the portfolio value is often
approximated as a quadratic function of percentage changes in the market variables. This
allows the first few moments of the probability distribution of the change in the portfolio
1
We are grateful to the editor Philippe Jorion for many suggestions that improved this paper.
2
value to be calculated analytically so that the required percentile of the distribution can be
estimated.2 An alternative approach to handling non-linearity is to use Monte Carlo
simulation. On each simulation trial daily changes in the market variables are sampled from
their multivariate distribution and the portfolio is revalued. This enables a complete
probability distribution for the daily change in the portfolio value to be determined.3
Many market variables have distributions with fatter tails than the normal distribution. This
has led some risk managers to use “historical simulation” rather than the model building
approach. Historical simulation involves creating a database consisting of the daily
movements in all market variables over a period of time. The first simulation trial assumes
that the percentage changes in the market variables are the same as on the first day
covered by the database; the second simulation trial assumes that they are the same as on
the second day; and so on. The change in the portfolio value is calculated for each
simulation trial and the required percentile of the probability distribution of this change is
estimated.4 As an example, suppose that 1,000 days of data are used and the 1 percentile
of the distribution is required. This would be estimated as the tenth worst change in the
portfolio value.
The advantage of the model building approach is that the underlying variance-covariance
matrix can be updated using an exponentially weighted moving average (EWMA) or
GARCH model.5 The disadvantage is that the market variables are assumed to be
conditionally multivariate normal. The model building approach takes no account of
skewness or kurtosis in the distributions of market variables and no account of nonlinear
correlations between market variables. Historical simulation, by contrast, has the
advantage that it accurately reflects the historical multivariate probability distribution for
the market variables. Its main disadvantage is that it incorporates no volatility updating.
Hull and White (1998) show how the assumption of multivariate normality in the model
building approach can be relaxed. Their approach allows any probability distribution to be
assumed for the unconditional daily changes in a market variable. A transformation is used
to convert the assumed distribution to a standard normal distribution. This transformation
is defined so that the X-percentile point of the assumed distribution is transformed to the
X-percentile of a standard normal distribution. The transformed market variables are
assumed to be multivariate normal. The approach was tested using nine years of data on
twelve different currencies and found to perform well.
2
When only two moments are calculated the distribution of the change in the portfolio value is assumed to
be normal. When three or more moments are calculated, the Cornish-Fisher expansion can be used to
estimate the required percentile.
3
Revaluing the complete portfolio on each simulation trial is usually not feasible because of the
computation time involved. One approach to speeding up calculations is assume that the change in the
portfolio value is a quadratic function of the change in the market variables.
4
As in the case of Monte Carlo simulation the quadratic approximation can be used as an alternative to a
full portfolio revaluation on each simulation trial.
5
For a discussion of these models see J.P. Morgan (1995) or Engle and Mezrich (1995)
3
The assumed distribution for each market variable in Hull and White (1998) can be chosen
in a variety of ways. One possibility is to select an appropriate standard distribution (for
example, a mixture of normals) and use maximum likelihood methods to find the best fit
parameters. Another possibility is to use the historical distribution. A third possibility is to
smooth the historical distribution; for example, by using a kernel estimator.6 The Hull and
White approach provides one way of bridging the gap between the model building and
historical simulation approaches. It shows how the model building approach can be
modified to incorporate some of the attractive features of the historical simulation
approach. In this paper we propose an alternative approach that allows volatility updating
to be incorporated into historical simulation.
2. Incorporating Volatility Updating Schemes into a Historical Simulation
The probability distribution of a market variable, when scaled by an estimate of its
volatility, is often found to be approximately stationary. This suggests that historical
simulation can be improved by taking account of the volatility changes experienced during
the period covered by the historical data. If the current volatility of a market variable is
1.5% per day and two months ago the volatility was only 1% per day, the data observed
two months ago understates the changes we expect to see now. On the other hand, if the
volatility was 2% per day two months ago the reverse is true.
We consider a portfolio dependent on a number of market variables and assume that the
variance of each market variable during the period covered by the historical data is
monitored using either a GARCH or EWMA model. We are interested in estimating VaR
for the portfolio at the end of day N-1 (i.e., for day N).
Define:
htj: the historical percentage change in variable j on day t of the period covered by the
historical sample (t<N)
σtj2: the historical GARCH/EWMA estimate of the daily variance of the percentage change
in variable j made for day t at the end of day t-1
The most recent GARCH/EWMA estimate of the daily variance is σNj2. This is the
estimate, made at the end of day N-1, of the variance of the percentage change in variable
j during day N. We assume that the probability distribution of htj/σtj is stationary. We
therefore replace each htj by htj* where
htj = σ N j
*
htj
σtj
(1)
6
An approach involving the use of historical simulation in conjunction with a kernel estimator is
suggested by Butler and Schachter (1998)
4
and set the t-th sample percentage change for variable j to htj* instead of htj.
This approach (which will be referred to as HW) is a straightforward extension of
traditional historical simulation (which will be referred to as HS). Instead of using the
actual historical percentage changes in market variables for the purposes of calculating
VaR, we use historical changes that have been adjusted to reflect the ratio of the current
daily volatility to the daily volatility at the time of the observation. Suppose that 20 days
ago the observed percentage change in a market variable was 1.6% and the daily volatility
was estimated to be 1%. If the daily volatility is now estimated to be 1.5%, the sample
percentage change calculated from the observation 20 days ago is 2.4%.
3. The BRW Approach
One of the ways in which risk managers attempt to allow for stochastic volatility is by
sampling more frequently from recent observations than from observations generated in
the distant past. Boudoukh, Richardson, and Whitelaw (1998) proposed one version of
this approach (which will be referred to as BRW). The weight given to the observation
n+1 days ago is λtimes the weight given to the observation n days ago where 0 < λ< 1.7
To determine a particular percentile of the probability distribution in BRW, it is necessary
to order the observations over the last N days and then, starting from the lowest one,
accumulate weights until the percentile is reached.8
We define a “5% tail event” as the occurrence of an observation that lies in the 5% tail of
the historical distribution and a “1% tail event” as the occurrence of an observation that
lies in the 1% tail of the historical distribution. Boudoukh, Richardson, and Whitelaw
(1998) indicate that, when regular historical simulation is used, 5% tail events do happen
approximately 5% of the time and 1% tail events do happen approximately 1% of the time.
However, there is significant “bunching”; that is, tail events tend to happen in close
succession rather than occurring randomly throughout the days covered by the data. An
attractive feature of the BRW approach is that it greatly reduces bunching.
BRW can be criticized on the grounds that it is an indirect and somewhat inefficient way
of allowing for stochastic volatility. In BRW (and all schemes that involve sampling more
frequently from recent observations) a short run sequence of abnormally large positive (or
negative) returns will markedly skew the predicted distribution to the right (or the left). In
BRW when λ=0.98, the most recent observation is assigned a probability of about 2% so
that a single large outcome is enough to generate this sort of skew. BRW and similar
7
Note that BRW use an EWMA approach to define the weights given to observations, but this is quite
different from the EWMA model for updating volatilities
8
Note that percentiles can be computed in a variety of ways. Suppose that a data set consists of the
numbers 1, 2, 3, and 4 (equally weighted). The definition we use throughout this paper implies 1, 2, 3,
and 4 are 25, 50, 75, and 100 percentiles, respectively, and the values for intermediate percentiles are
calculated using linear interpolation. An alternative definition (used by Microsoft’s Excel) implies that 1,
2, 3, and 4 are the 0, 33.33, 66.67, and 100 percentiles, respectively, and the values for intermediate
percentiles are calculated using linear interpolation.
5
schemes shorten the effective sampling period to capture the behavior of stochastic
volatility. Unfortunately in doing so they capture the stochastic behavior of all other
sample moments of the distribution.
4. Comparison of Approaches
We tested the three schemes (HS, BRW, and HW) using daily data on 12 different
exchange rates between January 4, 1988 and August 15, 1997 and five different stock
indices between July 11, 1988 and February 10, 1998. The currencies were the Australian
dollar (AUD), Belgian franc (BEF), Swiss franc (CHF), Deutschemark (DEM), Danish
krone (DKK), Spanish peseta (ESP), French franc (FRF), British pound (GBP), Italian lira
(ITL), Japanese yen (JPY), Dutch guilder (NLG), and Swedish krone (SEK). The stock
indices were the S&P 500, CAC-40, FT-SE 100, Nikkei 225, and Toronto Stock
Exchange 300. For each market variable, we had over 2,400 daily observations. For BRW
we used λ=0.98.9 In HW the daily variance was updated using the EWMA model
σ tj2 = ασ t2− 1, j + (1 − α )ht2− 1, j
(2)
with α = 0.94.10
For all three approaches and all market variables, a probability distribution of the daily
percentage change was estimated each day from the most recent 500 days of data. The 5
and 1 percentiles of the distribution were noted. For each market variable we define
indicator functions I(t) and J(t) for day t. I(t)=1 if the observed percentage change is less
than the 5 percentile on day t and zero otherwise; J(t)=1 if the observed percentage
change is less than the 1 percentile on day t and zero otherwise.
One issue in historical simulation is whether historical data should be adjusted to bring the
mean percentage change to zero. Consider for example the S&P 500. During the 500 days
ending February 10, 1998 the mean change was 0.09% per day. If we make no adjustment
to the historical data we are implicitly assuming that this is the expected change on
February 11, 1998. We tested each of the three approaches with and without a mean
adjustment. In the case of regular historical simulation, mean adjustment involved
subtracting the mean daily percentage change from each of the 500 observations prior to
estimating the 5% and 1% tails of the distribution. In the case of BRW, it involved
calculating a weighted mean percentage change and subtracting it from each observation.
In the case of HW, it involved calculating the mean of the normalized observations, htj/σtj,
and subtracting this mean from each normalized observation before multiplying by the
estimate of the current volatility, σNj.
9
Boudoukh, Richardson, and Whitelaw (1998) tested λ=0.9 9 and λ=0.9 7.
This is the model used by J.P. Morgan in their RiskMetrics database. See J.P. Morgan (1995)
10
6
Table 1 shows the percentage of days when tail events happen for exchange rates. Table 4
reports similar results for stock indices. If the tails of the estimated distributions were
unbiased, 5% of tail events would happen on 5% of the days and 1% of tail events would
happen on 1% of the days. Each result in Tables 1 and 4 is calculated from 1923
observations.11 The samples generating the empirical distributions are overlapping, but I(t)
and J(t) are each independent and identically distributed under the null hypothesis that the
tail of the distribution is unbiased. The standard deviation of the percentage of days when
tail events happen is therefore
p (1 − p )
n
where p is the probability of a tail event and n is the sample size. Asterisks in Tables 1 and
4 indicate situations where the hypothesis that the tail of the distribution is unbiased can be
rejected with 95% confidence. The tables show that the BRW method has a marked
tendency to understate 1% tail events. The results reported by Boudoukh, Richardson, and
Whitelaw (1998) show a similar phenomenon.
Boudoukh, Richardson, and Whitelaw (1998) propose a mean absolute percentage error
measure (MAPE) for measuring bunching. This is calculated as follows. For each period
of 100 consecutive days for which estimates are made, the absolute difference between the
actual number of tail events and the expected number of tail events is calculated. (For 5%
tail events the expected number of tail events is 5; for 1% tail events the expected number
of tail events is 1.) The measure is set equal to the mean of these absolute differences.
MAPE is a combined measure of both bias and bunching. The impact of a bias in the
measurement of tail events is clear. If the procedure for measuring tail events is biased so
that in every 100-day period we observe two 1%-tail events then MAPE = 1. To see how
the bunching component of the measure works consider the following example.
Suppose that there are 599 observations numbered 1 to 599. This allows us to compute
500 overlapping 100-day samples. If every 100th observation (observations 100, 200, 300,
400, and 500) are 1% tail events then every 100-day sample will contain exactly one 1%tail event and MAPE will be zero. Now suppose that the 1% tail events are bunched
together so that observations 100, 101, 300, 301, and 500 are tail events. Calculations
shows that there are 198 samples with no tail events (absolute error = 1), 104 with 1 tail
event (absolute error = 0), and 198 with 2 tail events (absolute error = 1). In this case
MAPE = 396/500 or 0.792.
The MAPE measure is similar to a standard deviation measure. If it were based the
difference between the observed number of tail events and the sample mean number of tail
events it would be even closer to a standard deviation measure. The definition used was
11
In the case of each currency the indicator functions can only be calculated from day 500 onward. This
explains why, although we started with over 2,400 observation per market variable, Tables 1 and 4 are
based on 1,923 observations per market variable.
7
chosen to maintain consistency with BRW. The measure is shown in Tables 2 and 5. For
each market variable the results are based on 1,823 different 100-day windows. Since the
windows are overlapping, standard tests of statistical significance cannot be used.
As an alternative measure of bunching we calculated the Ljung-Box statistic using the first
fifteen autocorrelations of the indicator functions, I(t) and J(t). In the case of 5% tail
events the autocorrelations were between the I(t); in the case of 1% tail events, they were
between the J(t). 12 The results are reported in Tables 3 and 6. A value of the statistic less
than 25 indicates that zero autocorrelation cannot be rejected at the 95% level. The
numbers for each market variable are based on 1,923 observations on the indicator
functions.
Tables 2, 3, 4, and 6 show that BRW and HW both produce big improvements over HS as
far as bunching is concerned. On average HW performs better than BRW. Tables 3 and 6
show that, for 5% tails, the hypothesis of zero autocorrelation cannot be rejected for 12
out of the 17 market variables considered when both HW (no mean adjustment) is used
and BRW (no mean adjustment) is used. For 1% tails, the hypothesis of zero
autocorrelation cannot be rejected for 16 of the market variables in the case of HW (no
mean adjustment) and for 11 of the market variables in the case of BRW (no mean
adjustment).
Overall the results with mean adjustment are similar to those without mean adjustment.
Most of the rest of the paper will focus on the results obtained without mean adjustment.
5. Capital Utilization
Define P as the 1 percentile of daily changes in a market variable. The regulatory risk
capital for an investment of $1 in a long position in the market variable is three times the
10-day 99% VaR or
− 3 10P
Figure 1 illustrates the three approaches for calculating P by showing the regulatory
capital that would be required for a long position of $1 in DEM. The DEM exchange rate
is also shown. As illustrated by the figure the capital is significantly more variable under
BRW and HW than under HS.
12
The Ljung-Box statistic is
15
m∑ w k ηk2
k =1
where m is the number of observations, ηk is the autocorrelation with a time lag of k days, and
wk = (m-2)/(m-k).
8
In the HS method the risk capital is unchanged for long periods of time due to the length
of the window. The risk capital is affected by large observations that appear in, and drop
off from, the window. An interesting aspect of BRW, illustrated in Figure 1, is that the
capital tends to increase to a new level for a period of time and then drop off sharply. To
understand the reason for this, consider a day (day d) where there is a sharp decrease in
the value of a portfolio, worse than anything seen on the previous 500 days. Assuming no
worse observation occurs subsequently, when λ=0.98 this increases the capital for exactly
35 subsequent days. The reason is as follows. On day d+1 the observation for day d is
given a weight of about 0.02 so that the 1 percentile of the distribution of daily returns
equals the day d observation. Between day 1 and day 35 the weight given to day d remains
above 1% so that the 1 percentile continues to equal the day d observation. On day 36 the
weight given to the observation is just below 1% and the next worst observation starts to
influences the 1 percentile point. BRW can be contrasted with HW where capital
requirements are determined by the most recent estimate of volatility and are therefore
much more responsive to recent observations.
For long positions in a single foreign currency we found that the capital required under
BRW is on average 11.0% less than under HS and that the capital required under HW is
on average 7.8% less than under HS. For long positions in a stock index we found that the
capital required under BRW is on average 0.2% higher than under HS and the capital
required under HW is on average 6.7% higher than under HS. Clearly BRW scores high
marks if the objective is to minimize capital. However, this is hardly surprising. Tables 1
and 4 show that BRW’s 1% tail are not extreme enough. A consistent result from our data
is that there is a greater than 1% chance of an observation being in BRW’s 1% tail.
Of course, the objective should not be to choose the method that minimizes capital
requirements. We contend that the best method is the one that, for a given average
investment of capital, maximizes the protection against losses. Define:
A:
CAVE:
Ct:
Ltn:
Average capital required under regular historical simulation
Average capital required under an alternative scheme
Capital required on day t under the alternative scheme
Losses incurred on the n days following day t under alternative scheme
The ratio Ltn/Ct is the proportion of the capital required to cover the losses (if any) during
the n days following day t under the alternative scheme. The extreme values of this
expression measure the chances of financial difficulties being experienced when the
alternative scheme is used. However, it does not take account of the average amount of
capital used by the alternative scheme. We propose that the measure
Ltn C AVE
(3)
Ct A
be used for choosing between schemes. This is the proportion of capital that would be
required on day t to cover losses in the subsequent n days under the alternative scheme if a
constant multiplier is applied to the capital each day so that it is the same on average as
9
the capital used under regular historical simulation. We will refer to this as the “capital
utilization ratio”. The extreme values of the capital utilization ratio provide a measure of
capital adequacy.
We calculated the 99.5 percentile and 99 percentile of the probability distribution of the
capital utilization ratio for investments in each of the currencies and each of the stock
indices for n=1 and n=10. The average of the results for each currency are shown in Table
7 and the average of the results for each stock index are shown in Table 8. The results
indicate that for currencies HW and BRW provide better capital utilization than HS. HW
performs better than BRW and the mean adjustment appears to improve the performance
of HW slightly. For stock indices the results are less clear. HW performs slightly better
than HS for a 1-day time horizon, but worse for a 10-day time horizon. BRW performs
worse than either HS or HW for a 1-day time horizon, and about the same as HW for a
10-day time horizon.
6. Summary
This paper shows how a volatility updating scheme such as GARCH can be used in
conjunction with historical simulation for calculating value at risk. Risk managers
sometimes attempt to allow for stochastic volatility by sampling from recent observations
more frequently than from those generated in the distant past. One such approach, BRW,
involves applying weights that decline exponentially to the observations. The approach we
propose is more direct. It involves adjusting observations to reflect the difference between
the volatility at the time of the observation and the current volatility.
We compare our approach with BRW. We use approximately 9 years of daily data on 12
different exchange rates and 5 different stock indices. Our approach appears to provide
better 1 percentile estimates of daily returns and is as good, if not better, at eliminating the
bunching of tail events.
We have proposed a new way of assessing of the effectiveness of a scheme for calculating
VaR. It is designed to test of the proportion of the capital likely to be used in extreme
situations. We find that our method performs better than BRW. It is superior to regular
historical simulation for investments in currencies. For investments in stock indices the
results are mixed.
10
References
Boudoukh, J, M. Richardson, and R. Whitelaw, “The Best of Both Worlds,” RISK, May
1998, pp 64-67.
Butler, J. S. and B. Schachter, “Estimating Value at Risk with a Precision Measure by
Combining Kernel Estimation with Historical Simulation” Review of Derivatives
Research, 1998, 1, pp. 371-390.
Engle, R.F., and J. Mezrich “Grappling with GARCH,” RISK, September 1995, pp.112117.
Hull, J. and A. White “Value at Risk when Daily Changes in Market Variables are not
Normally Distributed,” Journal of Derivatives, Spring 1998, pp 9-19.
Jorion, P., Value at Risk: The New Benchmark for Controlling Market Risk, Chicago:
Irwin, 1997.
J.P. Morgan, RiskMetrics Technical Manual, New York, J.P. Morgan Bank, 1995.
11
Figure 1
Capital required under three methods for a long position of $1 in DEM
HS: Regular Historical Simulation using 500 most recent observations
BRW: Historical Simulation Giving Weights that Decline Exponentially to 500 Most
Recent Observations
HW: Historical Simulation with Adjustment for Observations for Volatility Changes
0.8
0.7
0.6
0.5
HS
BRW
0.4
HW
FX Rate
0.3
0.2
0.1
Jun-97
Dec-96
Jun-96
Dec-95
Jun-95
Dec-94
Jun-94
Dec-93
Jun-93
Dec-92
Jun-92
Dec-91
Jun-91
Dec-90
Jun-90
Dec-89
0
12
Table 1
Percentage of Time Change in Exchange Rate Is Within 5% and 1% Tails of Estimated
Distribution Based Using Different Approaches (Based on 1923 Observations). Asterisk
Indicates That the Hypothesis of Unbiasedness Can be Rejected with 95% Confidence.
HS: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
BRW: Distribution Estimated by Giving Weights that Decline Exponentially to 500 most
Recent Observations
HW: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
After They Have Been Adjusted for Volatility Changes
5%
Tail
AUD
BEF
CHF
DEM
DKK
ESP
FRF
GBP
ITL
JPY
NLG
SEK
AVE
HS
No Mean Adj
4.05
4.73
4.68
4.57
4.68
4.78
4.73
4.83
4.78
4.73
4.68
5.41
4.72
HS
Mean Adj
4.21
4.57
4.57
4.57
4.37
4.83
4.47
4.68
4.78
4.47
4.68
5.25
4.62*
BRW
No Mean Adj
4.94
4.89
5.25
5.04
4.99
4.99
5.20
5.04
5.15
5.20
5.20
5.35
5.10
BRW
Mean Adj
4.89
4.57
4.89
4.73
4.52
4.78
5.09
4.78
5.15
4.73
4.94
5.15
4.85
HW
No Mean Adj
4.78
5.09
5.09
4.78
4.89
5.25
4.78
4.68
5.20
5.30
5.09
5.09
5.00
HW
Mean Adj
4.68
4.99
4.78
4.73
4.57
5.20
4.68
4.68
4.89
4.83
4.94
4.83
4.82
1%
Tail
AUD
BEF
CHF
DEM
DKK
ESP
FRF
GBP
ITL
JPY
NLG
SEK
AVE
HS
No Mean Adj
0.78
0.99
1.04
0.94
0.78
0.94
0.99
0.94
0.99
0.83
0.88
1.14
0.94
HS
Mean Adj
0.78
0.94
0.94
0.83
0.78
0.94
0.94
0.94
0.99
0.83
0.88
1.09
0.91
BRW
No Mean Adj
1.40
1.56*
1.35
1.51*
1.40
1.72*
1.35
1.30
1.40
1.72*
1.35
1.51*
1.46*
BRW
Mean Adj
1.40
1.35
1.30
1.25
1.40
1.72*
1.14
1.14
1.25
1.46*
1.20
1.40
1.33*
HW
No Mean Adj
1.04
0.94
0.88
1.09
1.04
0.94
1.09
1.14
0.88
0.83
1.04
1.04
1.00
HW
Mean Adj
0.99
0.88
0.94
1.09
0.99
0.94
0.99
1.14
0.88
0.78
1.04
0.94
0.97
13
Table 2
Mean Absolute Error in Percentage of 5% and 1% Tail Events for Exchange Rates in 100
consecutive days.
(Results Based on a Total of 1,823 100-day windows)
HS: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
BRW: Distribution Estimated by Giving Weights that Decline Exponentially to 500 most
Recent Observations
HW: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
After They Have Been Adjusted for Volatility Changes
5%
HS
Tail No Mean Adj
2.42
AUD
3.15
BEF
3.07
CHF
3.47
DEM
3.48
DKK
3.21
ESP
3.20
FRF
3.02
GBP
3.50
ITL
2.90
JPY
3.56
NLG
3.79
SEK
3.23
AVE
HS
Mean Adj
2.56
3.12
3.06
3.47
3.17
3.21
3.14
2.98
3.50
3.02
3.49
3.71
3.20
BRW
No Mean Adj
1.50
1.52
1.75
1.58
1.88
2.10
1.87
1.66
2.02
1.86
1.74
1.69
1.76
BRW
Mean Adj
1.65
1.76
1.78
1.63
1.90
2.21
1.88
1.45
1.96
1.62
1.69
1.68
1.77
HW
No Mean Adj
1.53
1.54
1.55
1.50
1.65
1.69
1.54
1.59
1.72
1.54
1.73
1.82
1.62
HW
Mean Adj
1.69
1.47
1.42
1.40
1.59
1.62
1.54
1.59
1.82
1.56
1.59
1.71
1.58
1%
HS
Tail No Mean Adj
0.73
AUD
1.09
BEF
1.36
CHF
1.24
DEM
0.95
DKK
0.91
ESP
1.01
FRF
0.91
GBP
1.22
ITL
1.10
JPY
1.19
NLG
0.99
SEK
1.06
AVE
HS
Mean Adj
0.73
1.09
1.28
1.15
0.95
0.91
0.96
0.91
1.22
1.10
1.19
1.05
1.05
BRW
No Mean Adj
0.77
0.98
0.90
0.91
0.91
1.04
0.78
0.89
0.77
0.98
0.95
0.77
0.89
BRW
Mean Adj
0.77
0.72
0.99
0.77
0.90
1.05
0.63
0.75
0.71
0.73
0.90
0.76
0.81
HW
No Mean Adj
0.72
0.48
0.55
0.69
0.79
0.55
0.69
0.70
0.65
0.72
0.68
0.80
0.67
HW
Mean Adj
0.73
0.53
0.51
0.69
0.73
0.55
0.69
0.70
0.65
0.72
0.68
0.73
0.66
14
Table 3
Ljung-Box Statistic for Autocorrelations Between Tail Events in Exchange Rates.
Results Based on Autocorrelations with Lags of Between 1 and 15 Days for Indicator
Function Which Equals 1 If Tail Event Happens and Zero Otherwise. Zero autocorrelation
cannot be rejected with 95% confidence when statistic is less than 25.
HS: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
BRW: Distribution Estimated by Giving Weights that Decline Exponentially to 500 most
Recent Observations
HW: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
After They Have Been Adjusted for Volatility Changes
5%
HS
Tail No Mean Adj
33.5
AUD
85.6
BEF
52.5
CHF
104.5
DEM
112.3
DKK
89.5
ESP
108.6
FRF
86.5
GBP
119.6
ITL
43.9
JPY
87.0
NLG
78.8
SEK
83.5
AVE
HS
Mean Adj
32.5
88.2
47.6
104.5
77.9
85.7
88.2
87.7
120.5
46.5
71.3
73.8
77.0
BRW
No Mean Adj
20.5
39.2
13.0
24.9
41.9
22.6
22.5
23.6
20.7
13.1
19.2
26.6
24.0
BRW
Mean Adj
18.5
41.6
19.6
21.3
36.1
25.9
25.6
22.9
21.0
10.3
22.6
27.2
24.4
HW
No Mean Adj
27.2
21.8
21.2
13.3
26.6
13.1
20.8
19.5
21.3
14.4
13.2
10.8
18.6
HW
Mean Adj
25.3
25.6
24.0
16.2
21.0
17.4
17.1
19.5
21.1
10.4
19.5
7.9
18.7
1%
HS
Tail No Mean Adj
8.5
AUD
53.1
BEF
153.4
CHF
79.2
DEM
21.8
DKK
67.3
ESP
16.5
FRF
50.8
GBP
134.4
ITL
59.1
JPY
73.3
NLG
83.0
SEK
66.7
AVE
HS
Mean Adj
8.5
60.4
147.8
56.7
21.8
67.3
14.6
50.8
134.4
59.1
73.3
92.5
65.6
BRW
No Mean Adj
7.9
21.7
26.3
18.3
33.5
28.4
13.8
46.6
20.1
10.1
29.1
8.6
22.0
BRW
Mean Adj
7.9
22.4
37.3
15.6
39.7
27.6
18.0
52.6
17.1
9.4
20.2
10.5
23.2
HW
No Mean Adj
6.1
6.6
7.4
42.4
14.7
6.6
10.8
7.9
12.6
8.1
11.8
9.0
12.0
HW
Mean Adj
6.3
7.1
6.9
42.4
9.7
6.6
9.7
7.9
12.6
9.0
11.8
10.6
11.7
15
Table 4
Percentage of Time Change in Stock Index Is Within 5% and 1% Tails of Estimated
Distribution Based Using Different Approaches (Based on 1923 Observations). Asterisk
Indicates That the Hypothesis of Unbiasedness Can be Rejected with 95% Confidence.
HS: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
BRW: Distribution Estimated by Giving Weights that Decline Exponentially to 500 most
Recent Observations
HW: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
After They Have Been Adjusted for Volatility Changes
5%
HS
Tail No Mean Adj
5.67
S&P
5.61
CAC
5.61
FTSE
5.51
NIKKEI
5.51
TSE
5.58*
AVE
HS
Mean Adj
5.09
5.41
5.25
5.77
4.94
5.29
BRW
No Mean Adj
5.30
5.51
5.61
5.41
5.77
5.52*
BRW
Mean Adj
4.63
5.25
5.09
5.93
4.94
5.17
HW
No Mean Adj
5.04
5.15
5.09
5.09
4.78
5.03
HW
Mean Adj
4.57
5.15
4.42
5.30
4.31
4.75
1%
HS
Tail No Mean Adj
1.40
S&P
1.51*
CAC
1.20
FTSE
1.25
NIKKEI
1.14
TSE
1.30*
AVE
HS
Mean Adj
1.20
1.30
1.04
1.30
1.04
1.17
BRW
No Mean Adj
1.46*
1.51*
1.40
1.35
1.20
1.38*
BRW
Mean Adj
1.35
1.56*
1.30
1.66*
0.99
1.37*
HW
No Mean Adj
0.83
1.09
1.04
0.73
0.88
0.91
HW
Mean Adj
0.83
1.09
1.04
0.78
0.78
0.90
16
Table 5
Mean Absolute Error in Percentage of 5% and 1% Tail Events for Stock Indices in 100
consecutive days.
(Results Based on a Total of 1,823 100-day windows)
HS: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
BRW: Distribution Estimated by Giving Weights that Decline Exponentially to 500 most
Recent Observations
HW: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
After They Have Been Adjusted for Volatility Changes
5%
HS
Tail No Mean Adj
2.68
S&P
2.99
CAC
3.23
FTSE
3.80
NIKKEI
2.71
TSE
3.08
AVE
HS
Mean Adj
2.57
2.88
3.29
3.83
2.33
2.98
BRW
No Mean Adj
1.62
1.61
1.88
2.19
1.86
1.83
BRW
Mean Adj
1.48
1.53
1.67
2.81
1.49
1.79
HW
No Mean Adj
1.52
1.66
1.83
2.22
1.58
1.76
HW
Mean Adj
1.47
1.69
1.49
2.29
1.60
1.71
1%
HS
Tail No Mean Adj
1.15
S&P
1.14
CAC
0.99
FTSE
1.21
NIKKEI
0.89
TSE
1.08
AVE
HS
Mean Adj
0.98
0.99
0.83
1.22
0.87
0.98
BRW
No Mean Adj
0.72
0.79
0.70
0.85
0.72
0.75
BRW
Mean Adj
0.61
0.77
0.67
1.20
0.59
0.77
HW
No Mean Adj
0.63
0.69
0.58
0.67
0.49
0.61
HW
Mean Adj
0.63
0.69
0.58
0.61
0.42
0.59
17
Table 6
Ljung-Box Statistic for Autocorrelations Between Tail Events in Exchange Rates.
Results Based on Autocorrelations with Lags of Between 1 and 15 Days for Indicator
Function Which Equals 1 If Tail Event Happens and Zero Otherwise. Zero autocorrelation
cannot be rejected with 95% confidence when statistic is less than 25.
HS: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
BRW: Distribution Estimated by Giving Weights that Decline Exponentially to 500 most
Recent Observations
HW: Distribution Estimated by Giving Equal Weights to 500 Most Recent Observations
After They Have Been Adjusted for Volatility Changes
5%
HS
Tail No Mean Adj
36.5
S&P
137.9
CAC
100.1
FTSE
319.6
NIKKEI
104.6
TSE
139.7
AVE
HS
Mean Adj
42.2
132.3
104.0
293.7
98.0
134.0
BRW
No Mean Adj
22.9
36.3
15.2
60.2
21.7
31.3
BRW
Mean Adj
14.0
55.9
26.0
81.9
17.8
39.1
HW
No Mean Adj
12.7
48.5
37.1
57.4
16.6
34.4
HW
Mean Adj
10.7
50.0
17.3
48.2
17.6
28.8
1%
HS
Tail No Mean Adj
24.6
S&P
254.4
CAC
26.8
FTSE
124.4
NIKKEI
52.7
TSE
96.6
AVE
HS
Mean Adj
21.8
206.8
33.2
112.0
43.0
83.3
BRW
No Mean Adj
30.4
24.2
7.3
12.4
11.0
17.1
BRW
Mean Adj
29.2
22.2
7.4
37.2
13.1
21.8
HW
No Mean Adj
7.7
29.8
3.2
17.4
7.1
13.0
HW
Mean Adj
7.7
29.8
3.2
15.1
1.8
11.5
18
Table 7
Percentiles of the Capital Utilization Ratio for Investments in Currencies.
The Capital Utilization Ratio Measures the Proportion of the Capital Used to Cover
Losses During the Specified Time Period. Results are the Averages of Those Obtained
from Investments in Each of 12 Different Currencies.
Time period
HS
(days)
Percentile No Mean Adj
1
99.5
12.54
1
99.0
10.35
10
99.5
38.49
10
99.0
32.38
HS
Mean Adj
12.44
10.33
38.25
32.16
BRW
No Mean Adj
12.08
10.50
38.12
32.33
BRW
Mean Adj
11.93
10.29
37.50
32.48
HW
No Mean Adj
11.71
9.87
38.05
32.12
HW
Mean Adj
11.66
9.79
37.61
31.88
Table 8
Percentiles of the Capital Utilization Ratio for Investments in Stock Indices.
The Capital Utilization Ratio Measures the Proportion of the Capital Used to Cover
Losses During the Specified Time Period. Results are the Averages of Those Obtained
from Investments in Each of 5 Different Stock Indices
Time Period
(days)
1
1
10
10
Percentile
99.5
99.0
99.5
99.0
HS
No Mean Adj
13.68
11.26
41.14
33.61
HS
Mean Adj
13.52
11.31
40.56
33.26
BRW
No Mean Adj
14.36
11.92
43.28
36.17
BRW
Mean Adj
14.56
11.97
43.18
35.95
HW
No Mean Adj
13.24
11.17
43.40
36.16
19
HW
Mean Adj
13.30
11.06
43.00
35.93